Calculus Examples

$\int \frac{x^{4} + 8}{x^{3}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{4} + 8) {(x^{3})}^{- 1} d x$

Step 1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (x^{4} + 8) x^{3 \cdot - 1} d x$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int (x^{4} + 8) x^{- 3} d x$

Step 2

Multiply $(x^{4} + 8) x^{- 3}$ .

$\int x^{4} x^{- 3} + 8 x^{- 3} d x$

Step 3

Simplify.

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Step 3.1

Multiply $x^{4}$ by $x^{- 3}$ by adding the exponents.

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Step 3.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{4 - 3} + 8 x^{- 3} d x$

Step 3.1.2

Subtract $3$ from $4$ .

$\int x^{1} + 8 x^{- 3} d x$

Step 3.2

Simplify $x^{1}$ .

$\int x + 8 x^{- 3} d x$

Step 4

Split the single integral into multiple integrals.

$\int x d x + \int 8 x^{- 3} d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2} + C + \int 8 x^{- 3} d x$

Step 6

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$\frac{1}{2} x^{2} + C + 8 \int x^{- 3} d x$

Step 7

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$\frac{1}{2} x^{2} + C + 8 (- \frac{1}{2} x^{- 2} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$\frac{1}{2} x^{2} + C + 8 (- \frac{x^{- 2}}{2} + C)$

Step 8.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} x^{2} + C + 8 (- \frac{1}{2 x^{2}} + C)$

Step 8.2

Simplify.

$\frac{1}{2} x^{2} + 8 (- \frac{1}{2 x^{2}}) + C$

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $8$ .

$\frac{1}{2} x^{2} - 8 \frac{1}{2 x^{2}} + C$

Step 8.3.2

Combine $- 8$ and $\frac{1}{2 x^{2}}$ .

$\frac{1}{2} x^{2} + \frac{- 8}{2 x^{2}} + C$

Step 8.3.3

Cancel the common factor of $- 8$ and $2$ .

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Step 8.3.3.1

Factor $2$ out of $- 8$ .

$\frac{1}{2} x^{2} + \frac{2 \cdot - 4}{2 x^{2}} + C$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $2$ out of $2 x^{2}$ .

$\frac{1}{2} x^{2} + \frac{2 \cdot - 4}{2 (x^{2})} + C$

Step 8.3.3.2.2

Cancel the common factor.

$\frac{1}{2} x^{2} + \frac{2 \cdot - 4}{2 x^{2}} + C$

Step 8.3.3.2.3

Rewrite the expression.

$\frac{1}{2} x^{2} + \frac{- 4}{x^{2}} + C$

Step 8.3.4

Move the negative in front of the fraction.

$\frac{1}{2} x^{2} - \frac{4}{x^{2}} + C$